On the unicity of the formal theory of categoriestetrapharmakon.github.io/stuff/fix_slides.pdfFosco Loregian Unicity December 4, 2018 - ULB17/36 LetK bea2-category;aYonedastructureonK

On the unicity of the formal theory of categories

Fosco Loregian

Max Planck Institute for Mathematics

December 4, 2018 - ULB

Fosco Loregian Unicity December 4, 2018 - ULB 1 / 36

All I’m about to say is joint work with Ivan Di Liberti (MasarykUNIversity, CZ).

A few questions remain open, but we’ll be on arXiv soon!Every comment or advice is welcome.


All I’m about to say is joint work with Ivan Di Liberti (MasarykUNIversity, CZ).A few questions remain open, but we’ll be on arXiv soon!

Every comment or advice is welcome.


All I’m about to say is joint work with Ivan Di Liberti (MasarykUNIversity, CZ).A few questions remain open, but we’ll be on arXiv soon!Every comment or advice is welcome.


Plan of the talk

Fast and painless intro to formal category theory

Setting the stage: bits of 2-category theory I’ll need along thediscussionthe current state of art: Yoneda lemma VS bimodulesMain theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”Examples:

Representable Yoneda structureTwo-sided Yoneda structureIsbell duality and its generalizations

An open and motivating problem: derivators.


Plan of the talk

Fast and painless intro to formal category theorySetting the stage: bits of 2-category theory I’ll need along thediscussion

the current state of art: Yoneda lemma VS bimodulesMain theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”Examples:




Plan of the talk

Fast and painless intro to formal category theorySetting the stage: bits of 2-category theory I’ll need along thediscussionthe current state of art: Yoneda lemma VS bimodules

Main theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”Examples:




Plan of the talk

Fast and painless intro to formal category theorySetting the stage: bits of 2-category theory I’ll need along thediscussionthe current state of art: Yoneda lemma VS bimodulesMain theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”

Examples:Representable Yoneda structureTwo-sided Yoneda structureIsbell duality and its generalizations



Plan of the talk

Fast and painless intro to formal category theorySetting the stage: bits of 2-category theory I’ll need along thediscussionthe current state of art: Yoneda lemma VS bimodulesMain theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”Examples:




Plan of the talk

Fast and painless intro to formal category theorySetting the stage: bits of 2-category theory I’ll need along thediscussionthe current state of art: Yoneda lemma VS bimodulesMain theorem: “Yoneda structures are yosegi, yosegis are proarrowequipments.”Examples:




Introduction


Formal category theory is a branch of 2-category theoryit serves to axiomatize the structural part of category theory

In the words of John Gray

1.«The purpose of category theory is to try to describe certain generalaspects of the structure of mathematics. Since category theory is also partof mathematics, this categorical type of description should apply to it aswell as to other parts of mathematics.»

2.«The basic idea is that the category of small categories, Cat, is a“2-category with properties”; one should attempt to identify thoseproperties that enable one to do the “structural parts of category theory”.»












1. reads as “category theory is a theory we can interpret in differentcontexts”; these contexts are 2-categories.

2. reads as “our task is unravel the properties enabling to treat an abstract2-category K as if it were Cat”.

cf. topos theory: treat E as if it were Set;cf. categorical algebra: treat A as if it were AlgpTq;cf. homological algebra: treat C as if it were ChpAq;


1. reads as “category theory is a theory we can interpret in differentcontexts”; these contexts are 2-categories.2. reads as “our task is unravel the properties enabling to treat an abstract2-category K as if it were Cat”.



1. reads as “category theory is a theory we can interpret in differentcontexts”; these contexts are 2-categories.2. reads as “our task is unravel the properties enabling to treat an abstract2-category K as if it were Cat”.



Perhaps surprisingly though, the bare structure of a 2-category does notsuffice to embody “all” category theory in a formal way.

Sure you can do many things:adjunctions: pairs of 1-cells f : X Ô Y : g

monads: endo-1-cells t : AÑ A

Kan extensionsfibrations


Perhaps surprisingly though, the bare structure of a 2-category does notsuffice to embody “all” category theory in a formal way.

Sure you can do many things:adjunctions: pairs of 1-cells f : X Ô Y : g

monads: endo-1-cells t : AÑ A

Kan extensionsfibrations


but many other important things are missing:

a “pointwise” formula to compute Kan extensionsequivalent characterization of adjunctions: Y pf , 1q – X p1, gq

Y) the Yoneda lemma (“category theory is the Yoneda lemma”)E) a calculus of modules (“category theory is the theory of multi-object

monoids”)

IdeaTake a 2-category K, and take 1. and 2. very seriously.


but many other important things are missing:

a “pointwise” formula to compute Kan extensionsequivalent characterization of adjunctions: Y pf , 1q – X p1, gq

Y) the Yoneda lemma (“category theory is the Yoneda lemma”)E) a calculus of modules (“category theory is the theory of multi-object

monoids”)

IdeaTake a 2-category K, and take 1. and 2. very seriously.


(Y): Yoneda structures

Category theory is the class of corollaries of Yoneda lemma.

A Yoneda structure imposes on K enough structure so that every“small” object A has a “Yoneda embedding” yA : AÑ PA.Formally encode the Yoneda lemma in universal properties of certaindiagrams

Af

yA

~~PA

" χf

BBpf ,1q

oo

representing f -nerves.



Category theory is the class of corollaries of Yoneda lemma.A Yoneda structure imposes on K enough structure so that every“small” object A has a “Yoneda embedding” yA : AÑ PA.

Formally encode the Yoneda lemma in universal properties of certaindiagrams

Af

yA

~~PA

" χf

BBpf ,1q

oo




Category theory is the class of corollaries of Yoneda lemma.A Yoneda structure imposes on K enough structure so that every“small” object A has a “Yoneda embedding” yA : AÑ PA.Formally encode the Yoneda lemma in universal properties of certaindiagrams

Af

yA

~~PA

" χf

BBpf ,1q

oo



(E): proarrow equipments

Category theory is the theory of multi-object monoids.

And monoids are well-known for their irresistible tendency to act onobjects!This leads to the notion of C -D-bimodule as a functorC_ ˆ D Ñ Set. All such bimodules live in a bicategory Mod , andthere is an embedding

p : Cat ãÑ Mod

such that every f P Cat has a right adjoint when embedded in Mod .Every such p : KÑ K equips K with proarrows.



Category theory is the theory of multi-object monoids.And monoids are well-known for their irresistible tendency to act onobjects!

This leads to the notion of C -D-bimodule as a functorC_ ˆ D Ñ Set. All such bimodules live in a bicategory Mod , andthere is an embedding

p : Cat ãÑ Mod




Category theory is the theory of multi-object monoids.And monoids are well-known for their irresistible tendency to act onobjects!This leads to the notion of C -D-bimodule as a functorC_ ˆ D Ñ Set. All such bimodules live in a bicategory Mod , andthere is an embedding

p : Cat ãÑ Mod

such that every f P Cat has a right adjoint when embedded in Mod .

Every such p : KÑ K equips K with proarrows.



Category theory is the theory of multi-object monoids.And monoids are well-known for their irresistible tendency to act onobjects!This leads to the notion of C -D-bimodule as a functorC_ ˆ D Ñ Set. All such bimodules live in a bicategory Mod , andthere is an embedding

p : Cat ãÑ Mod



The main theorem of this talk is an equivalence between a certain class ofYoneda structures and a certain class of proarrow equipments:

yosegiboxes

(

! cocompleteYoneda

structures

)

Yonedaequipments

(

What allows to pass from YS to E is the notion of a yosegi, a certainspecial kind of 2-monad.



yosegiboxes

(

! cocompleteYoneda

structures

)

Yonedaequipments

(




yosegiboxes

(

! cocompleteYoneda

structures

)

Yonedaequipments

(



Preliminaries


2-dimensional universality

The main 2-dimensional universal constructions we are interested in

A ñη

g

f // B

C

Lang f

;;

Lang f Ñ hf Ñ hg

Liftg f Ñ hf Ñ gh C

g

B

f//

Liftg f;;

Añη

A

ñεg

f // B

C

Rang f

;;

hg Ñ fh Ñ Rang f

h Ñ Riftg fgh Ñ f C

g

B

f//

Riftg f;;

A

ñε

pointwise. . . . . . absolute




A ñη

g

f // B

C

Lang f

;;

Lang f Ñ hf Ñ hg


g

B

f//

Liftg f;;

Añη

A

ñεg

f // B

C

Rang f

;;

hg Ñ fh Ñ Rang f


g

B

f//

Riftg f;;

A

ñε

pointwise. . .

. . . absolute




A ñη

g

f // B

C

Lang f

;;

Lang f Ñ hf Ñ hg


g

B

f//

Liftg f;;

Añη

A

ñεg

f // B

C

Rang f

;;

hg Ñ fh Ñ Rang f


g

B

f//

Riftg f;;

A

ñε

pointwise. . . . . . absolute


Proposition (formal description of adjoints)

The following conditions are equivalent, for a pair of 1-cells f : A Ô B : g :f % g with unit η and counit ε;The pair xg , ηy exhibits the absolute Lan of 1 along f

The pair xg , ηy ehhibits the Lan of 1 along f , and f preserves it.


A bit of coend calculus:

Proposition (ninja Yoneda lemma)

For every presheaf F : C Ñ Set, it holds

F pX q –

ÂFX hompA,X q F pX q –

ˆ A

FAˆ hompX ,Aq

naturally in X .

Proposition (pointwise Kan extensions)

For a diagram C GÐÝ A F

ÝÑ B, it holds

LanGF pX q –

ˆ A

hompGA,X q ˆ FA

naturally in X .


A bit of coend calculus:

Proposition (ninja Yoneda lemma)

For every presheaf F : C Ñ Set, it holds

F pX q –

ÂFX hompA,X q F pX q –

ˆ A

FAˆ hompX ,Aq

naturally in X .

Proposition (pointwise Kan extensions)

For a diagram C GÐÝ A F

ÝÑ B, it holds

LanGF pX q –

ˆ A

hompGA,X q ˆ FA

naturally in X .


Yoneda structures


Let K be a 2-category; a Yoneda structure on K is made of

Yoneda data

An ideal of admissible arrows Jidentity morphisms in the ideal determine admissible objects.

a family of morphisms yA : AÑ PA one for each admissible object Abeware! PA is only syntactic sugar. It has no meaning whatsoever.

A family of triangles

A

f

yA

~~PA

" χf

BBpf ,1q

oo

filled by a 2-cell χ : yA Ñ Bpf , f q “: Bpf , 1q ¨ f .


Let K be a 2-category; a Yoneda structure on K is made ofYoneda data




A

f

yA

~~PA

" χf

BBpf ,1q

oo







A

f

yA

~~PA

" χf

BBpf ,1q

oo







A

f

yA

~~PA

" χf

BBpf ,1q

oo







A

f

yA

~~PA

" χf

BBpf ,1q

oo



subject to Yoneda axioms

The pair xBpf , 1q, χf y exhibits the pointwise left extension Lanf yA.The pair xf , χf y exhibits the absolute left lifting LiftBpf ,1qyA (shortly,f %yA Bpf , 1q).The pair x1PA, 1yAy exhibits the pointwise left extension LanyAyA(shortly, ‘the Yoneda embedding is dense’).Given a pair of composable 1-cells A f

ÝÑ BgÝÑ C , the pasting of 2-cells

A PA

B PB

C

f

yA

yB

g

Pf

Cpg,1q

χyB f

χg

exhibits the pointwise extension Langf yA “ C pgf , 1q (shortly, P is afunctor).


subject to Yoneda axiomsThe pair xBpf , 1q, χf y exhibits the pointwise left extension Lanf yA.

The pair xf , χf y exhibits the absolute left lifting LiftBpf ,1qyA (shortly,f %yA Bpf , 1q).The pair x1PA, 1yAy exhibits the pointwise left extension LanyAyA(shortly, ‘the Yoneda embedding is dense’).Given a pair of composable 1-cells A f


A PA

B PB

C

f

yA

yB

g

Pf

Cpg,1q

χyB f

χg



subject to Yoneda axiomsThe pair xBpf , 1q, χf y exhibits the pointwise left extension Lanf yA.The pair xf , χf y exhibits the absolute left lifting LiftBpf ,1qyA (shortly,f %yA Bpf , 1q).

The pair x1PA, 1yAy exhibits the pointwise left extension LanyAyA(shortly, ‘the Yoneda embedding is dense’).Given a pair of composable 1-cells A f


A PA

B PB

C

f

yA

yB

g

Pf

Cpg,1q

χyB f

χg



subject to Yoneda axiomsThe pair xBpf , 1q, χf y exhibits the pointwise left extension Lanf yA.The pair xf , χf y exhibits the absolute left lifting LiftBpf ,1qyA (shortly,f %yA Bpf , 1q).The pair x1PA, 1yAy exhibits the pointwise left extension LanyAyA(shortly, ‘the Yoneda embedding is dense’).

Given a pair of composable 1-cells A fÝÑ B

gÝÑ C , the pasting of 2-cells

A PA

B PB

C

f

yA

yB

g

Pf

Cpg,1q

χyB f

χg



subject to Yoneda axiomsThe pair xBpf , 1q, χf y exhibits the pointwise left extension Lanf yA.The pair xf , χf y exhibits the absolute left lifting LiftBpf ,1qyA (shortly,f %yA Bpf , 1q).The pair x1PA, 1yAy exhibits the pointwise left extension LanyAyA(shortly, ‘the Yoneda embedding is dense’).Given a pair of composable 1-cells A f


A PA

B PB

C

f

yA

yB

g

Pf

Cpg,1q

χyB f

χg



In Cat

The ideal J is made by functors f : AÑ B such that arrows faÑ bform a set for every a, b P A,B ; admissible objects are made by locallysmall categories;

yA is of course the Yoneda embedding AÑ rAop,Sets;χf is obtained from the action of f on arrows.


In Cat

The ideal J is made by functors f : AÑ B such that arrows faÑ bform a set for every a, b P A,B ; admissible objects are made by locallysmall categories;yA is of course the Yoneda embedding AÑ rAop,Sets;

χf is obtained from the action of f on arrows.


In Cat

The ideal J is made by functors f : AÑ B such that arrows faÑ bform a set for every a, b P A,B ; admissible objects are made by locallysmall categories;yA is of course the Yoneda embedding AÑ rAop,Sets;χf is obtained from the action of f on arrows.


In Cat

Axiom 1

Bpf , 1q “ λb.λa. homBpfa, bq exhibits with χf the pointwise left Kanextension of yA along f : AÑ B .

It is enough to check that

rB,PAspNf ,G q –

ˆbPApBpf , bq,Gbq

–

âb

SetpBpfa, bq,G pbqpaqq

– G pfaqpaq

rA,PAspyA,G ¨ f q –

âPApyApaq,G pfaqq

– G pfaqpaq.


In Cat

Axiom 1

Bpf , 1q “ λb.λa. homBpfa, bq exhibits with χf the pointwise left Kanextension of yA along f : AÑ B .


rB,PAspNf ,G q –

ˆbPApBpf , bq,Gbq

–

âb

SetpBpfa, bq,G pbqpaqq

– G pfaqpaq

rA,PAspyA,G ¨ f q –

âPApyApaq,G pfaqq

– G pfaqpaq.


In Cat

Axiom 2

The pair xf , χf y exhibits a relative adjunction f %yABpf , 1q.


rA,PAs`

yA,Nf ¨ g˘

–

â1

rAop,Sets`

yAa1,Nf ¨ gpa

1q˘

–

â1

rAop,Sets`

yAa1,Bpf´, ga1q

˘

–

â1

Bpfa1, ga1q

– rA,Bspf , gq

absoluteness can be checkd by hand.


In Cat

Axiom 2

The pair xf , χf y exhibits a relative adjunction f %yABpf , 1q.


rA,PAs`

yA,Nf ¨ g˘

–

â1

rAop,Sets`

yAa1,Nf ¨ gpa

1q˘

–

â1

rAop,Sets`

yAa1,Bpf´, ga1q

˘

–

â1

Bpfa1, ga1q

– rA,Bspf , gq

absoluteness can be checkd by hand.


In Cat

Axiom 3-4They admit a rephrasing as P is a functor, in that Ppidq – id andPpgf q – Pf ¨ Pg .

This is pretty obvious (although the isomorphisms

3) rPA,PAsp1,Hq – rA,PAspyA,H ¨ yAq;4) C pgf , 1q “ Langf yA – LanyB f yA ˝ LangyB “ PBpyB f , 1q ¨ C pg , 1q

can be checked directly).


In Cat

Axiom 3-4They admit a rephrasing as P is a functor, in that Ppidq – id andPpgf q – Pf ¨ Pg .

This is pretty obvious (although the isomorphisms

3) rPA,PAsp1,Hq – rA,PAspyA,H ¨ yAq;4) C pgf , 1q “ Langf yA – LanyB f yA ˝ LangyB “ PBpyB f , 1q ¨ C pg , 1q

can be checked directly).


All Cat-like 2-categories carry Yoneda structures in the obvious way:V-Cat (enriched categories) with the enriched Yoneda embeddings;internal categories in a finitely complete category A;the 2-category of pseudofunctors AÑ Cat for a small bicategory A;. . .


Proarrowequipments


Definition (proarrow equipment)

A 2-functor p : KÑ K equips K with proarrows ifp is the identity on objects and locally fully faithful;p is such that for each 1-cell f : AÑ B in K ppf q has a right adjointin K.

Examples: the embedding Cat ãÑ Mod into the category ofbimodules/profunctors. The embedding of Catc (Cauchy completecategories) in geometric morphisms of presheaf toposes.

Very simple definition; a bit complicated to follow the literature. Recentlyframed into the more natural environment of (hyper)virtual doublecategories of (Shulman-Crutwell).












QuestionHow do these framework relate? It is reasonable to expect they do (they’reboth ways to encode a calculus of profunctors; yA : AÑ PA is the (mateof the )identity profunctor).


Strategy1 in a Yoneda structure P is a functor by axiom 3-4; the correspondence

A ÞÑ PA is monad-like

2 the Kleisli category of P equips K with proarrows via the free part ofits free-forget adjunction!

3 if only it was possible to go back to a Yoneda structure. . .

Turns out 1. is almost true; 2. is true; 3. is true if we restrict to so-calledYoneda equipments where p has additional properties.



A ÞÑ PA is monad-like2 the Kleisli category of P equips K with proarrows via the free part of

its free-forget adjunction!

3 if only it was possible to go back to a Yoneda structure. . .





its free-forget adjunction!3 if only it was possible to go back to a Yoneda structure. . .





its free-forget adjunction!3 if only it was possible to go back to a Yoneda structure. . .



The presheaf construction sending A ÞÑ rAop,Sets and f : AÑ B to theinverse image P˚f : PB Ñ PA is such that

1 for each f : AÑ B there is an adjunction P !pf q % P˚pf q;2 P ! is a relative monad (relative to the inclusion Cat Ă CAT);3 the monad P ! is a KZ-doctrine.



1 for each f : AÑ B there is an adjunction P !pf q % P˚pf q;

2 P ! is a relative monad (relative to the inclusion Cat Ă CAT);3 the monad P ! is a KZ-doctrine.



1 for each f : AÑ B there is an adjunction P !pf q % P˚pf q;2 P ! is a relative monad (relative to the inclusion Cat Ă CAT);

3 the monad P ! is a KZ-doctrine.



1 for each f : AÑ B there is an adjunction P !pf q % P˚pf q;2 P ! is a relative monad (relative to the inclusion Cat Ă CAT);3 the monad P ! is a KZ-doctrine.


Definition (yosegi box)

Let j : A Ă K and P : AÑ K with the same properties, and such thatmoreover

1 the unit ηA of the monad P ! is fully faithful;

2 The cell P˚ηA exists (the codomain of ηA might be non-admissible!)We call P a (j-relative) yosegi box.a

aYosegi-zaiku (寄木細工) is a kind of marquetry featuring elaborate inlaid andmosaic designs; the defining properties of such a KZ-doctrine are tightly linked, rich ofpeculiar adornments.




1 the unit ηA of the monad P ! is fully faithful;2 The cell P˚ηA exists (the codomain of ηA might be non-admissible!)

We call P a (j-relative) yosegi box.a






We call P a (j-relative) yosegi box.

a






We call P a (j-relative) yosegi box.a



The main theorem

Let j : A Ă K the inclusion of a sub-2-category; the following conditionsare equivalent.

1 K has a Yoneda structure with presheaf construction P, and admissibleobjects those of A, moreover, every PA is a cocomplete object;

2 the inclusion j : A Ă K admits a Yoneda equipment p %j u;3 The inclusion j : A Ă K extends to a yosegi box.


The main theorem





The main theorem



2 the inclusion j : A Ă K admits a Yoneda equipment p %j u;

3 The inclusion j : A Ă K extends to a yosegi box.


The main theorem





a Yoneda equipment is a triangle

A

K Mñ

pj

u

such thatp is a proarrow equipment a la Wood;p is the j-relative left adjoint of u : MÑ K;the relative monad up generated by the relative adjunction, is laxidempotent with unit η : j ñ up.

the inclusion j extends to a yosegi if there is a relative lax-idempotent2-monad P : AÑ K with relative unit j ñ P such that for each 1-cellf : AÑ B , the 1-cell Ppf q has a left adjoint P !f .


a Yoneda equipment is a triangle

A

K Mñ

pj

u

such thatp is a proarrow equipment a la Wood;p is the j-relative left adjoint of u : MÑ K;the relative monad up generated by the relative adjunction, is laxidempotent with unit η : j ñ up.

the inclusion j extends to a yosegi if there is a relative lax-idempotent2-monad P : AÑ K with relative unit j ñ P such that for each 1-cellf : AÑ B , the 1-cell Ppf q has a left adjoint P !f .


Proof is technical, not enlightening, but amazingly all the pieces fittogether as in a perfectly built carpentry work.

The key assumption here is that A ‘embeds well’ into K via j (as soon asone of the conditions above is true, j is very near to be dense).

the Kleisli category of a yosegi P : AÑ K equips its domain A withproarrows;every cocomplete Yoneda structure has a yosegi as presheafconstruction;Yoneda equipments contain enough information to recover a Yonedastructure.








the Kleisli category of a yosegi P : AÑ K equips its domain A withproarrows;

every cocomplete Yoneda structure has a yosegi as presheafconstruction;Yoneda equipments contain enough information to recover a Yonedastructure.




the Kleisli category of a yosegi P : AÑ K equips its domain A withproarrows;every cocomplete Yoneda structure has a yosegi as presheafconstruction;

Yoneda equipments contain enough information to recover a Yonedastructure.






Applications


Examples

Often, P is representable, because K is cartesian closed:P : A ÞÑ rAop,Ωs; this is the case for many K’s having a dualityinvolution p qop, and entails that P has an adjoint

P7 % P

In Cat, P7 is the “contravariant presheaf construction”, sending A torA,Setsop.


This self-duality of P is tightly linked to the possibility to instantiatethe formal version of Isbell duality: there is an adjunction

Ay 7

A

yA

~~PA

O // P7ASpec

oo

where O “ LanyApy7

Aq and Spec “ Lany 7

ApyAq.

The pair pP7,Pq forms a two-sided Yoneda structure/yosegi: we havecopresheaf constructions working as free completion:P7 : A ÞÑ rA,Setsop does precisely this in Cat; the axioms of a (left)Yoneda structure hold replacing left extensions and lifts with thecorresponding right versions. These two structures are compatible.


This self-duality of P is tightly linked to the possibility to instantiatethe formal version of Isbell duality: there is an adjunction

Ay 7

A

yA

~~PA

O // P7ASpec

oo

where O “ LanyApy7

Aq and Spec “ Lany 7

ApyAq.

The pair pP7,Pq forms a two-sided Yoneda structure/yosegi: we havecopresheaf constructions working as free completion:P7 : A ÞÑ rA,Setsop does precisely this in Cat; the axioms of a (left)Yoneda structure hold replacing left extensions and lifts with thecorresponding right versions. These two structures are compatible.


An enticing conjecture is that

On the 2-category of derivators there is a Yoneda structure that keepstrack of the homotopy theory of derivators.

The yosegi it generates is represented by the derivator of spaces, which hasthe universal property of the “free cocompletion of the point”. A sketch ofthe precise construction: consider the following diagram

cat Cat8 Der A sSetNA HopsSetNAq

rcatop, cats ypAq

ν

y

The arrow y is the Yoneda embedding of small categories into smallprederivators; taking the Yoneda extension of the functor ν : A ÞÑ sSetNA

now we get a functor from pDer (lowercase p = small prederivators) to(presentable) 8-categories, whose homotopy categories are thus derivators.




The yosegi it generates is represented by the derivator of spaces, which hasthe universal property of the “free cocompletion of the point”.

A sketch ofthe precise construction: consider the following diagram


rcatop, cats ypAq

ν

y








rcatop, cats ypAq

ν

y








rcatop, cats ypAq

ν

y




Documents

On the unicity of the formal theory of categoriestetrapharmakon.github.io/stuff/fix_slides.pdfFosco Loregian Unicity December 4, 2018 - ULB17/36 LetK bea2-category;aYonedastructureonK